1. Field
Apparatus and methods for performing quantum computing are provided. The systems and methods involve the use of superconducting circuitry and, more specifically, the use of devices for quantum computation.
2. Description of the Related Art
A Turing machine is a theoretical computing system, described in 1936 by Alan Turing. A Turing machine that can efficiently simulate any other Turing machine is called a Universal Turing Machine (UTM). The Church-Turing thesis states that any practical computing model has either the equivalent or a subset of the capabilities of a UTM.
A quantum computer is any physical system that harnesses one or more quantum effects to perform a computation. A quantum computer that can efficiently simulate any other quantum computer is called a Universal Quantum Computer (UQC).
In 1981 Richard P. Feynman proposed that quantum computers could be used to solve certain computational problems more efficiently than a UTM and therefore invalidate the Church-Turing thesis. See e.g., Feynman R. P., “Simulating Physics with Computers”, International Journal of Theoretical Physics, Vol. 21 (1982) pp. 467-488. For example, Feynman noted that a quantum computer could be used to simulate certain other quantum systems, allowing exponentially faster calculation of certain properties of the simulated quantum system than is possible using a UTM.
Approaches to Quantum Computation
There are several general approaches to the design and operation of quantum computers. One such approach is the “circuit model” of quantum computation. In this approach, qubits are acted upon by sequences of logical gates that are the compiled representation of an algorithm. Circuit model quantum computers have several serious barriers to practical implementation. In the circuit model, it is required that qubits remain coherent over time periods much longer than the single-gate time. This requirement arises because circuit model quantum computers require operations that are collectively called quantum error correction in order to operate. Quantum error correction cannot be performed without the circuit model quantum computer's qubits being capable of maintaining quantum coherence over time periods on the order of 1,000 times the single-gate time. Much research has been focused on developing qubits with coherence sufficient to form the basic information units of circuit model quantum computers. See e.g., Shor, P. W. “Introduction to Quantum Algorithms”, arXiv.org:quant-ph/0005003 (2001), pp. 1-27. The art is still hampered by an inability to increase the coherence of qubits to acceptable levels for designing and operating practical circuit model quantum computers.
An example of the circuit model is shown in FIG. 2. Circuit 200 is an implementation of the quantum Fourier transform. The quantum Fourier transform is a useful procedure found in many quantum computing applications based on the circuit model. See, for example, United States Patent Publication 2003/0164490 A1, entitled “Optimization process for quantum computing process,” which is hereby incorporated by reference in its entirety. Time progresses from left to right, i.e., time step 201 precedes time step 202, and so forth. The four qubits in the quantum system described by FIG. 2 are indexed 0-3 from bottom to top. The state of qubit 0 at any given time step is represented by wire S0-S0″, the state of qubit 1 at any give time step is represented by S1-S1′, etc. In time step 201, a single-qubit unitary gate, A3, is applied to qubit 3. The next gate on wire S3-S3′ for qubit 3 is a two-qubit gate, B23, which is applied to qubits 2 and 3 at time step 202. In general the Ai gate (e.g., A3 as applied to qubit 3 at time step 201) is a HADAMARD gate applied on the ith qubit while the Bij gate (e.g., B23 which is applied to qubits 2 and 3 at time step 202) is a CPHASE gate coupling the ith and jth qubit. The application of unitary gates continues until states S0-S3 have been converted to S0′-S3′. After time step 210, more unitary gates can be applied to the qubits or the states of the qubits can be determined (e.g., by measurement).
Another approach to quantum computation, involves using the natural physical evolution of a system of coupled quantum systems as a computational system. This approach does not make critical use of quantum gates and circuits. Instead, starting from a known initial Hamiltonian, it relies upon the guided physical evolution of a system of coupled quantum systems wherein the problem to be solved has been encoded in the terms of the system's Hamiltonian, so that the final state of the system of coupled quantum systems contains information relating to the answer to the problem to be solved. This approach does not require long qubit coherence times. Examples of this type of approach include adiabatic quantum computation, cluster-state quantum computation, one-way quantum computation, quantum annealing and classical annealing, and are described, for example, in Farhi, E. et al., “Quantum Adiabatic Evolution Algorithms versus Stimulated Annealing” arXiv.org:quant-ph/0201031 (2002), pp 1-24.
As shown in FIG. 3, adiabatic quantum computing involves initializing a system, which encodes a problem to be solved, to an initial state. This initial state is described by an initial Hamiltonian H0. Then the system is migrated adiabatically to a final state described by Hamiltonian HP. The final state encodes a solution to the problem. The migration from H0 to HP follows an interpolation path described by function γ(t) that is continuous over the time interval zero to T, inclusive, and has a condition that the magnitude of initial Hamiltonian H0 is reduced to zero after time T. Here, T, refers to the time point at which the system reaches the state represented by the Hamiltonian HP. Optionally, the interpolation can traverse an extra Hamiltonian HE that can contain tunneling terms for some or all of the qubits represented by H0. The magnitude of extra Hamiltonian HE is described by a function δ(t) that is continuous over the time interval zero to T, inclusive, and is zero at the start (t=0) and end (t=T) of the interpolation while being non-zero at all or a portion of the times between t=0 and t=T.
Qubits
As mentioned previously, qubits can be used as fundamental units of information for a quantum computer. As with bits in UTMs, qubits can refer to at least two distinct quantities; a qubit can refer to the actual physical device in which information is stored, and it can also refer to the unit of information itself, abstracted away from its physical device.
Qubits generalize the concept of a classical digital bit. A classical information storage device can encode two discrete states, typically labeled “0” and “1”. Physically these two discrete states are represented by two different and distinguishable physical states of the classical information storage device, such as direction or magnitude of magnetic field, current, or voltage, where the quantity encoding the bit state behaves according to the laws of classical physics. A qubit also contains two discrete physical states, which can also be labeled “0” and “1”. Physically these two discrete states are represented by two different and distinguishable physical states of the quantum information storage device, such as direction or magnitude of magnetic field, current, or voltage, where the quantity encoding the bit state behaves according to the laws of quantum physics. If the physical quantity that stores these states behaves quantum mechanically, the device can additionally be placed in a superposition of 0 and 1. That is, the qubit can exist in both a “0” and “1” state at the same time, and so can perform a computation on both states simultaneously. In general, N qubits can be in a superposition of 2N states. Quantum algorithms make use of the superposition property to speed up some computations.
In standard notation, the basis states of a qubit are referred to as the |0 and |1 states. During quantum computation, the state of a qubit, in general, is a superposition of basis states so that the qubit has a nonzero probability of occupying the |0 basis state and a simultaneous nonzero probability of occupying the |1 basis state. Mathematically, a superposition of basis states means that the overall state of the qubit, which is denoted |Ψ, has the form |Ψ=a|0+b|1, where a and b are coefficients corresponding to the probabilities |a|2 and |b|2, respectively. The coefficients a and b each have real and imaginary components, which allows the phase of the qubit to be characterized. The quantum nature of a qubit is largely derived from its ability to exist in a coherent superposition of basis states and for the state of the qubit to have a phase. A qubit will retain this ability to exist as a coherent superposition of basis states when the qubit is sufficiently isolated from sources of decoherence.
To complete a computation using a qubit, the state of the qubit is measured (i.e., read out). Typically, when a measurement of the qubit is performed, the quantum nature of the qubit is temporarily lost and the superposition of basis states collapses to either the |0 basis state or the |1 basis state and thus regaining its similarity to a conventional bit. The actual state of the qubit after it has collapsed depends on the probabilities |a|2 and |b|2 immediately prior to the readout operation.
Superconducting Qubits
There are many different hardware and software approaches under consideration for use in quantum computers. One hardware approach uses integrated circuits formed of superconducting materials, such as aluminum or niobium. The technologies and processes involved in designing and fabricating superconducting integrated circuits are similar to those used for conventional integrated circuits.
Superconducting qubits are a type of superconducting device that can be included in a superconducting integrated circuit. Superconducting qubits can be separated into several categories depending on the physical property used to encode information. For example, they may be separated into charge, flux and phase devices, as discussed in, for example Makhlin et al., 2001, Reviews of Modern Physics 73, pp. 357-400. Charge devices store and manipulate information in the charge states of the device, where elementary charges consist of pairs of electrons called Cooper pairs. A Cooper pair has a charge of 2e and consists of two electrons bound together by, for example, a phonon interaction. See e.g., Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000), pp. 343-345. Flux devices store information in a variable related to the magnetic flux through some part of the device. Phase devices store information in a variable related to the difference in superconducting phase between two regions of the phase device. Recently, hybrid devices using two or more of charge, flux and phase degrees of freedom have been developed. See e.g., U.S. Pat. No. 6,838,694 and U.S. Patent Application No. 2005-0082519.
FIG. 1A illustrates a persistent current qubit 101. Persistent current qubit 101 comprises a loop 103 of superconducting material interrupted by Josephson junctions 101-1, 101-2, and 101-3. Josephson junctions are typically formed using standard fabrication processes, generally involving material deposition and lithography stages. See, e.g., Madou, Fundamentals of Microfabrication, Second Edition, CRC Press, 2002. Methods for fabricating Josephson junctions are well known and described in Ramos et al., 2001, IEEE Trans. App. Supercond. 11, 998, for example. Details specific to persistent current qubits can be found in C. H. van der Wal, 2001; J. B. Majer, 2002; and J. R. Butcher, 2002, all Theses in Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands; http://qt.tn.tudelft.nl; Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands. Common substrates include silicon, silicon oxide, or sapphire, for example. Josephson junctions can also include insulating materials such as aluminum oxide, for example. Exemplary superconducting materials useful for forming superconducting loop 103 are aluminum and niobium. The Josephson junctions have cross-sectional sizes ranging from about 10 nanometers (nm) to about 10 micrometers (μm). One or more of the Josephson junctions 101 has parameters, such as the size of the junction, the junction surface area, the Josephson energy or the charging energy that differ from the other Josephson junctions in the qubit.
The difference between any two Josephson junctions in the persistent current qubit is characterized by a coefficient, termed α, which typically ranges from about 0.5 to about 1.3. In some instances, the term α for a pair of Josephson junctions in the persistent current qubit is the ratio of the critical current between the two Josephson junctions in the pair. The critical current of a Josephson junction is the minimum current through the junction at which the junction is no longer superconducting. That is, below the critical current, the junction is superconducting whereas above the critical current, the junction is not superconducting. Thus, for example, the term α for junctions 101-1 and 101-2 is defined as the ratio between the critical current of junction 101-1 and the critical current of junction 101-2.
Referring to FIG. 1A, a bias source 110 is inductively coupled to persistent current qubit 101. Bias source 110 is used to thread a magnetic flux Φx through phase qubit 101 to provide control of the state of the phase qubit. In some instances, the persistent current qubit operates with a magnetic flux bias Φx ranging from about 0.2·Φ0 to about 0.8·Φ0, where Φ0 is the flux quantum. In some instances, the magnetic flux bias ranges from about 0.47·Φ0 to about 0.5·Φ0.
Persistent current qubit 101 has a two-dimensional potential with respect to the phase across Josephson junctions 101-1, 101-2, and 101-3. In some instances, persistent current qubit 101 is biased with a magnetic flux Φx, such that the two-dimensional potential profile includes regions of local energy minima, where the local energy minima are separated from each other by small energy barriers and are separated from other regions by large energy barriers. In some instances, this potential has the shape of double well potential 100B (FIG. 1B), which includes a left well 160-0 and a right well 160-1. In such instances, left well 160-0 can represent clockwise (102-0) circulating supercurrent in the phase qubit 101 and right well 160-1 can represent counter-clockwise (102-1) circulating supercurrent in persistent current qubit 101 of FIG. 1A.
When wells 160-0 and 160-1 are at or near degeneracy, meaning that they are at the same or nearly the same energy potential as illustrated in FIG. 1B, the quantum state of persistent current qubit 101 becomes a coherent superposition of the phase or basis states and device can be operated as a phase qubit. The point at or near degeneracy is herein referred to as the point of computational operation of the persistent current. During computational operation of the persistent current qubit, the charge of the qubit is fixed leading to uncertainty in the phase basis and delocalization of the phase states of the qubit. Controllable quantum effects can then be used to process the information stored in those phase states according to the rules of quantum mechanics. This makes the persistent current qubit robust against charge noise and thereby prolongs the time under which the qubit can be maintained in a coherent superposition of basis states.
Computational Complexity Theory
In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given computational problem. This cost is usually measured in terms of abstract parameters such as time and space, called computational resources. Time represents the number of steps required to solve a problem and space represents the quantity of information storage required or how much memory is required.
Computational complexity theory classifies computational problems into complexity classes. The number of complexity classes is ever changing, as new ones are defined and existing ones merge through the contributions of computer scientists. The complexity classes of decision problems include:                1. P—The complexity class containing decision problems that can be solved by a deterministic UTM using a polynomial amount of computation time;        2. NP (“Non-deterministic Polynomial time”)—The set of decision problems solvable in polynomial time on a non-deterministic UTM. Equivalently, it is the set of problems that can be “verified” by a deterministic UTM in polynomial time;        3. NP-hard (Non-deterministic UTM Polynomial-time hard)—The class of decision problems that contains all problems H, such that for every decision problem L in NP there exists a polynomial-time many-one reduction to H. Informally, this class can be described as containing the decision problems that are at least as hard as any problem in NP;        4. NP-complete—A decision problem C is NP-complete if it is complete for NP, meaning that:                    (a) it is in NP and            (b) it is NP-hard,                        i.e., every other problem in NP is reducible to it. “Reducible” means that for every problem L, there is a polynomial-time many-one reduction, a deterministic algorithm which transforms instances I∈L into instances c∈C, such that the answer to c is YES if and only if the answer to I is YES. To prove that an NP problem A is in fact an NP-complete problem it is sufficient to show that an already known NP-complete problem reduces to A.        
Decision problems have binary outcomes. Problems in NP are computation problems for which there exists a polynomial time verification. That is, it takes no more than polynomial time (class P) in the size of the problem to verify a potential solution. It may take more than polynomial time, however, to find a potential solution. NP-hard problems are at least as hard as any problem in NP.
Optimization problems are problems for which one or more objective functions are minimized or maximized over a set of variables, sometimes subject to a set of constraints. For example, the Traveling Salesman Problem (“TSP”) is an optimization problem where an objective function representing, for example, distance or cost, must be optimized to find an itinerary, which is encoded in a set of variables representing the optimized solution to the problem. For example, given a list of locations, the problem may consist of finding the shortest route that visits all locations exactly once. Other examples of optimization problems include Maximum Independent Set, integer programming, constraint optimization, factoring, prediction modeling, and k-SAT. These problems are abstractions of many real-world optimization problems, such as operations research, financial portfolio selection, scheduling, supply management, circuit design, and travel route optimization. Many large-scale decision-based optimization problems are NP-hard. See e.g., “A High-Level Look at Optimization: Past, Present, and Future” e-Optimization.com, 2000.
Simulation problems typically deal with the simulation of one system by another system, usually over a period of time. For example, computer simulations can be made of business processes, ecological habitats, protein folding, molecular ground states, quantum systems, and the like. Such problems often include many different entities with complex inter-relationships and behavioral rules. In Feynman it was suggested that a quantum system could be used to simulate some physical systems more efficiently than a UTM.
Many optimization and simulation problems are not solvable using UTMs. Because of this limitation, there is need in the art for computational devices capable of solving computational problems beyond the scope of UTMs. In the field of protein folding, for example, grid computing systems and supercomputers have been used to try to simulate large protein systems. See Shirts et al., 2000, Science 290, pp. 1903-1904, and Allen et al., 2001, IBM Systems Journal 40, p. 310. The NEOS solver is an online network solver for optimization problems, where a user submits an optimization problem, selects an algorithm to solve it, and then a central server directs the problem to a computer in the network capable of running the selected algorithm. See e.g., Dolan et al., 2002, SIAM News Vol. 35, p. 6. Other digital computer-based systems and methods for solving optimization problems can be found, for example, in Fourer et al., 2001, Interfaces 31, pp. 130-150. All these methods are limited, however, by the fact they utilize digital computers, which are UTMs, and accordingly, are subject to the limits of classical computing that impose unfavorable scaling between problem size and solution time.